# §25.10 Zeros

## §25.10(i) Distribution

The product representation (25.2.11) implies $\mathop{\zeta\/}\nolimits\!\left(s\right)\neq 0$ for $\Re{s}>1$. Also, $\mathop{\zeta\/}\nolimits\!\left(s\right)\neq 0$ for $\Re{s}=1$, a property first established in Hadamard (1896) and de la Vallée Poussin (1896a, b) in the proof of the prime number theorem (25.16.3). The functional equation (25.4.1) implies $\mathop{\zeta\/}\nolimits\!\left(-2n\right)=0$ for $n=1,2,3,\dots$. These are called the trivial zeros. Except for the trivial zeros, $\mathop{\zeta\/}\nolimits\!\left(s\right)\neq 0$ for $\Re{s}\leq 0$. In the region $0<\Re{s}<1$, called the critical strip, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ has infinitely many zeros, distributed symmetrically about the real axis and about the critical line $\Re{s}=\frac{1}{2}$. The Riemann hypothesis states that all nontrivial zeros lie on this line.

Calculations relating to the zeros on the critical line make use of the real-valued function

 25.10.1 $Z(t)=\mathop{\exp\/}\nolimits\!\left(i\vartheta(t)\right)\mathop{\zeta\/}% \nolimits\!\left(\tfrac{1}{2}+it\right),$

where

 25.10.2 $\vartheta(t)\equiv\mathop{\mathrm{ph}\/}\nolimits\mathop{\Gamma\/}\nolimits\!% \left(\tfrac{1}{4}+\tfrac{1}{2}it\right)-\tfrac{1}{2}t\mathop{\ln\/}\nolimits\pi$

is chosen to make $Z(t)$ real, and $\mathop{\mathrm{ph}\/}\nolimits\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{4}+% \frac{1}{2}it\right)$ assumes its principal value. Because $|Z(t)|=|\mathop{\zeta\/}\nolimits\!\left(\frac{1}{2}+it\right)|$, $Z(t)$ vanishes at the zeros of $\mathop{\zeta\/}\nolimits\!\left(\frac{1}{2}+it\right)$, which can be separated by observing sign changes of $Z(t)$. Because $Z(t)$ changes sign infinitely often, $\mathop{\zeta\/}\nolimits\!\left(\frac{1}{2}+it\right)$ has infinitely many zeros with $t$ real.

## §25.10(ii) Riemann–Siegel Formula

Riemann developed a method for counting the total number $N(T)$ of zeros of $\mathop{\zeta\/}\nolimits\!\left(s\right)$ in that portion of the critical strip with $0. By comparing $N(T)$ with the number of sign changes of $Z(t)$ we can decide whether $\mathop{\zeta\/}\nolimits\!\left(s\right)$ has any zeros off the line in this region. Sign changes of $Z(t)$ are determined by multiplying (25.9.3) by $\mathop{\exp\/}\nolimits\!\left(i\vartheta(t)\right)$ to obtain the Riemann–Siegel formula:

 25.10.3 $Z(t)=2\sum_{n=1}^{m}\frac{\mathop{\cos\/}\nolimits\!\left(\vartheta(t)-t% \mathop{\ln\/}\nolimits n\right)}{n^{1/2}}+R(t),$

where $R(t)=\mathop{O\/}\nolimits\!\left(t^{-1/4}\right)$ as $t\to\infty$.

The error term $R(t)$ can be expressed as an asymptotic series that begins

 25.10.4 $R(t)=(-1)^{m-1}\left(\frac{2\pi}{t}\right)^{1/4}\frac{\mathop{\cos\/}\nolimits% \!\left(t-(2m+1)\sqrt{2\pi t}-\frac{1}{8}\pi\right)}{\mathop{\cos\/}\nolimits% \!\left(\sqrt{2\pi t}\right)}+\mathop{O\/}\nolimits\!\left(t^{-3/4}\right).$

Riemann also developed a technique for determining further terms. Calculations based on the Riemann–Siegel formula reveal that the first ten billion zeros of $\mathop{\zeta\/}\nolimits\!\left(s\right)$ in the critical strip are on the critical line (van de Lune et al. (1986)). More than one-third of all the zeros in the critical strip lie on the critical line (Levinson (1974)).

For further information on the Riemann–Siegel expansion see Berry (1995).